You bought six numbers at your local hardware store.
The numbers are 0, 1, 2, 3, 4, 5.
a) How many 6 digit house numbers would be even?
(note: 0351 is not considered a house number as you
only use each number once).
First we find the number of ways to choose the 1st
and 6th (last) digits.
Case 1. We choose 2 or 4 for the first digit
We choose the first digit 2 ways, 2 or 4
We choose the last digit 2 ways, 0 and whichever
of 2,4 we did not choose for the first digit.
That's 2*2 = 4 ways for case 1.
Case 2. We choose 1,3, or 5 for the 1st digit
We choose the 1st digit 3 ways, 1,3 or 5
We choose the 6th (last) digit 3 ways, 0, 2, or 4
choose for the first digit.
That's 3*3 = 9 ways for case 2.
Total number of ways to choose 1st and 6th (last) digits =
4+9 = 13 ways to choose 1st and 6th (last) digits.
For each of those 13 ways to choose the 1st and 6th digits,
we can choose the 2nd digit any of the 4 remaining ways.
we can choose the 3rd digit any of the 3 remaining ways.
we can choose the 4th digit either of the 2 remaining ways.
we can choose the 5th digit only 1 remaining way.
Answer: 13*4*3*2*1 = 312 ways.
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b) Using the same digits as above, how many 4 digit
numbers smaller than 3500 could you form if repetition
was not allowed? (note: 0351 is not considered a house
number).
This does not say anything about the number being even or odd,
so we will assume it can be either even or odd.
First we find the number of ways to choose the 1st
and 2nd digits.
Case 1. The first digit is 3
We choose the first digit as 3. That's 1 way.
We choose the 2nd digit as 0, 1, 2, or 4. That's 4 ways.
That's 1*4 = 4 ways for case 1.
Case 2. The first digit is 1 or 2
We choose the first digit as 1 or 2. That's 2 ways
We choose the 2nd digit as any of the 5 remaining digits.
That's 2*5 = 10 ways for case 2.
Total for cases 1 and 2 = 4+10 = 14
For each of those 14 ways to choose the 1st and 2nd digits,
we can choose the 3rd digit any of the 4 remaining ways.
we can choose the 4th digit any of the 3 remaining ways.
Answer: 14*4*3 = 168 ways
Edwin